Multiplying and Dividing Integers — Sign Rules Explained

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After reading this page you will know the sign rules that govern every integer multiplication and division problem. You will be able to predict the sign of a product or quotient instantly, handle chains of three or more factors, and avoid the most common sign-related mistakes on tests and homework.

Theory

The two sign rules

Multiplication and division share the exact same sign rules. You only need to remember two things:

$\text{Same signs} \longrightarrow \text{positive result}$

$\text{Different signs} \longrightarrow \text{negative result}$

Here is the complete table:

Factor 1	Factor 2	Product / Quotient
$+$	$+$	$+$
$-$	$-$	$+$
$+$	$-$	$-$
$-$	$+$	$-$

Why does negative times negative equal positive?

Think of multiplication as repeated addition. $3 \times (-4)$ means "add $-4$ three times":

$(-4) + (-4) + (-4) = -12$

Now $(-3) \times (-4)$ means "take away $-4$ three times." Removing a debt is the same as gaining money:

$-(-4) - (-4) - (-4) = 4 + 4 + 4 = +12$

This is why two negatives produce a positive in multiplication.

Multiplying with absolute values

A reliable method is to ignore the signs, multiply the absolute values, then attach the correct sign:

$(-7) \times 6$

Step 1: Multiply absolute values: $7 \times 6 = 42$.

Step 2: Different signs ($-$ and $+$), so the result is negative.

Answer: $(-7) \times 6 = -42$

Division works the same way:

$(-36) \div (-9)$

Step 1: Divide absolute values: $36 \div 9 = 4$.

Step 2: Same signs ($-$ and $-$), so the result is positive.

Answer: $(-36) \div (-9) = +4$

Multiplying three or more integers

When you multiply three or more integers, count the negative factors:

• Even number of negatives $\longrightarrow$ positive product
• Odd number of negatives $\longrightarrow$ negative product

$(-2) \times (-3) \times (-5) = \;?$

Absolute-value product: $2 \times 3 \times 5 = 30$. There are three negative factors (odd), so the product is negative: $-30$.

Division as the inverse of multiplication

Since multiplication and division are inverse operations, the sign rules stay consistent:

$(-5) \times 8 = -40 \quad\Longleftrightarrow\quad (-40) \div (-5) = 8 \quad\text{and}\quad (-40) \div 8 = -5$

If you ever forget a division sign rule, think of the related multiplication fact.

Special cases with zero

Multiplying any integer by zero always gives zero, regardless of signs:

$(-15) \times 0 = 0$

Division by zero is undefined -- it has no answer. But zero divided by any nonzero integer is zero:

$0 \div (-7) = 0$

Worked Examples

Example 1: Multiplying two integers (easy)

Problem: Calculate $(-8) \times 5$.

Step 1: Multiply the absolute values. $8 \times 5 = 40$

Step 2: The signs are different ($-$ and $+$), so the result is negative.

Answer: $(-8) \times 5 = \mathbf{-40}$

Example 2: Dividing two negative integers (easy)

Problem: Calculate $(-54) \div (-6)$.

Step 1: Divide the absolute values. $54 \div 6 = 9$

Step 2: Both signs are negative (same signs), so the result is positive.

Answer: $(-54) \div (-6) = \mathbf{+9}$

Example 3: Three-factor multiplication (medium)

Problem: Calculate $(-4) \times (-3) \times 7$.

Step 1: Multiply the absolute values. $4 \times 3 \times 7 = 84$

Step 2: Count the negative factors. There are two negatives (even), so the product is positive.

Answer: $(-4) \times (-3) \times 7 = \mathbf{84}$

Example 4: Mixed multiplication and division (medium)

Problem: Simplify $(-60) \div 5 \times (-2)$.

Step 1: Work left to right. First, divide. $(-60) \div 5 = -12 \quad \text{(different signs: negative)}$

Step 2: Then multiply. $(-12) \times (-2) = +24 \quad \text{(same signs: positive)}$

Answer: $(-60) \div 5 \times (-2) = \mathbf{24}$

Example 5: Four-factor product (challenging)

Problem: Calculate $(-1) \times (-2) \times (-3) \times (-4)$.

Step 1: Multiply the absolute values. $1 \times 2 \times 3 \times 4 = 24$

Step 2: Count the negative factors. There are four negatives (even), so the product is positive.

Answer: $(-1) \times (-2) \times (-3) \times (-4) = \mathbf{+24}$

Bonus check: Multiply step by step. $(-1) \times (-2) = +2$ $2 \times (-3) = -6$ $(-6) \times (-4) = +24 \;\checkmark$

Common Mistakes

Mistake 1: Applying addition sign rules to multiplication

❌ $(-3) \times (-5) = -15$ (thinking "both negative, so the answer is negative")

✅ $(-3) \times (-5) = +15$

Why this matters: In addition, two negatives make a more negative result. In multiplication, two negatives cancel out and give a positive result. Mixing up these rules is one of the most common errors on integer tests.

Mistake 2: Forgetting to count all negative factors

❌ $(-2) \times (-3) \times (-4) = +24$ (only looking at the first two negatives)

✅ Count all three negatives (odd) $\to$ result is $-24$

Why this matters: When multiplying more than two integers, every negative factor matters. Count them all before deciding the sign.

Mistake 3: Dividing by zero

❌ $(-12) \div 0 = 0$

✅ $(-12) \div 0$ is undefined (no answer exists)

Why this matters: Zero divided by a number is zero, but dividing by zero is impossible. This distinction is crucial in algebra and higher math.

Practice Problems

Try these on your own before checking the answers:

1. $(-9) \times 7$
2. $(-42) \div (-7)$
3. $(-5) \times (-4) \times (-2)$
4. $80 \div (-10) \times (-3)$
5. $(-1) \times (-1) \times (-1) \times (-1) \times (-1)$

Click to see answers

1. $(-9) \times 7 = \mathbf{-63}$ (different signs: negative)
2. $(-42) \div (-7) = \mathbf{6}$ (same signs: positive)
3. $5 \times 4 \times 2 = 40$; three negatives (odd) $\to$ $\mathbf{-40}$
4. $80 \div (-10) = -8$; $(-8) \times (-3) = \mathbf{24}$ (same signs: positive)
5. Absolute values: $1$. Five negatives (odd) $\to$ $\mathbf{-1}$

Summary

• Same signs ($+ \times +$ or $- \times -$) always give a positive result.
• Different signs ($+ \times -$ or $- \times +$) always give a negative result.
• The sign rules for division are identical to those for multiplication.
• For three or more factors: count the negatives. Even count $=$ positive; odd count $=$ negative.
• Multiply or divide absolute values first, then attach the correct sign.

Related Topics

• Integers — Operations, Number Line, and Word Problems
• Adding and Subtracting Integers — Rules and Examples
• Integer Word Problems with Step-by-Step Solutions

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